Quintic threefold – Wikipedia

Posted on August 28, 2022 by lordneo

In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space

P4{displaystyle mathbb {P} ^{4}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef0f42e9d791c201ab1f11cfeaef6372d876364" aria-hidden="true" alt="{displaystyle mathbb {P} ^{4}}" width="1065.4" height="1152.1">$ . Non-singular quintic threefolds are Calabi–Yau manifolds.

The Hodge diamond of a non-singular quintic 3-fold is

Mathematician Robbert Dijkgraaf said “One number which every algebraic geometer knows is the number 2,875 because obviously, that is the number of lines on a quintic.”^[1]

Table of Contents

Definition[edit]

A quintic threefold is a special class of Calabi–Yau manifolds defined by a degree

5{displaystyle 5}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" aria-hidden="true" alt="5" width="500.5" height="936.9">$ projective variety in

P4{displaystyle mathbb {P} ^{4}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef0f42e9d791c201ab1f11cfeaef6372d876364" aria-hidden="true" alt="{displaystyle mathbb {P} ^{4}}" width="1065.4" height="1152.1">$ , or as a smooth variety resolving the singularities of another variety. As a set, a Calabi-Yau manifold is

X={x=[x0:x1:x2:x3:x4]∈CP4:p(x)=0}{displaystyle X={x=[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}]in mathbb {CP} ^{4}:p(x)=0}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4da81588faa99bb60eee1bcb2aca69e5332b6c" aria-hidden="true" alt="{displaystyle X={x=[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}]in mathbb {CP} ^{4}:p(x)=0}}" width="21653.9" height="1367.4">

where

p(x){displaystyle p(x)}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" aria-hidden="true" alt="p(x)" width="1893.5" height="1223.9">$ is a degree

5{displaystyle 5}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" aria-hidden="true" alt="5" width="500.5" height="936.9">$ homogeneous polynomial. One of the most studied examples is from the polynomial

p(x)=x05+x15+x25+x35+x45{displaystyle p(x)=x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8648b095ced9d48e63094574c023e197cf4b6240" aria-hidden="true" alt="{displaystyle p(x)=x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}}" width="13251.4" height="1367.4">

called a Fermat polynomial. Proving that such a polynomial defines a Calabi-Yau requires some more tools, like the Adjunction formula and conditions for smoothness.

Hypersurfaces in P⁴[edit]

Recall that a homogeneous polynomial

f∈Γ(P4,O(d)){displaystyle fin Gamma (mathbb {P} ^{4},{mathcal {O}}(d))}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4db679db368cf2355dc1f7c99dd47694c888c0a" aria-hidden="true" alt="{displaystyle fin Gamma (mathbb {P} ^{4},{mathcal {O}}(d))}" width="6787.6" height="1367.4">$ (where

O(d){displaystyle {mathcal {O}}(d)}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/610a8b689907ff816a0515f34ff08c1d3dc175d6" aria-hidden="true" alt="{displaystyle {mathcal {O}}(d)}" width="2099" height="1223.9">$ is the Serre-twist of the hyperplane line bundle) defines a projective variety, or projective scheme,

X{displaystyle X}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" aria-hidden="true" alt="X" width="852.5" height="936.9">$ , from the algebra

k[x0,…,x4](f){displaystyle {frac {k[x_{0},ldots ,x_{4}]}{(f)}}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99e477dc0669b5df2e51d0495f274cc6304e16d0" aria-hidden="true" alt="{displaystyle {frac {k[x_{0},ldots ,x_{4}]}{(f)}}}" width="5720.8" height="2802.6">

where

k{displaystyle k}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" aria-hidden="true" alt="k" width="521.5" height="936.9">$ is a field, such as

C{displaystyle mathbb {C} }

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" aria-hidden="true" alt="mathbb {C} " width="722.5" height="936.9">$ . Then, using the adjunction formula to compute its canonical bundle, we have

ΩX3=ωX=ωP4⊗O(d)≅O(−(4+1))⊗O(d)≅O(d−5){displaystyle {begin{aligned}Omega _{X}^{3}&=omega _{X}\&=omega _{mathbb {P} ^{4}}otimes {mathcal {O}}(d)\&cong {mathcal {O}}(-(4+1))otimes {mathcal {O}}(d)\&cong {mathcal {O}}(d-5)end{aligned}}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71719aff7052ce0617adb207fc0c03bde55db16e" aria-hidden="true" alt="{displaystyle {begin{aligned}Omega _{X}^{3}&amp;=omega _{X}\&amp;=omega _{mathbb {P} ^{4}}otimes {mathcal {O}}(d)\&amp;cong {mathcal {O}}(-(4+1))otimes {mathcal {O}}(d)\&amp;cong {mathcal {O}}(d-5)end{aligned}}}" width="11761.8" height="5529.4">

hence in order for the variety to be Calabi-Yau, meaning it has a trivial canonical bundle, its degree must be

5{displaystyle 5}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" aria-hidden="true" alt="5" width="500.5" height="936.9">$ . It is then a Calabi-Yau manifold if in addition this variety is smooth. This can be checked by looking at the zeros of the polynomials

∂0f,…,∂4f{displaystyle partial _{0}f,ldots ,partial _{4}f} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3a04d37fceb616d898405377abadecf93b3363" aria-hidden="true" alt="{displaystyle partial _{0}f,ldots ,partial _{4}f}" width="5301.3" height="1080.4">

and making sure the set

{x=[x0:⋯:x4]|f(x)=∂0f(x)=⋯=∂4f(x)=0}{displaystyle {x=[x_{0}:cdots :x_{4}]|f(x)=partial _{0}f(x)=cdots =partial _{4}f(x)=0}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590c63bebbe79f14d7359f4f2c8dca64343326d0" aria-hidden="true" alt="{displaystyle {x=[x_{0}:cdots :x_{4}]|f(x)=partial _{0}f(x)=cdots =partial _{4}f(x)=0}}" width="23322.5" height="1223.9">

is empty.

Examples[edit]

Fermat Quintic[edit]

One of the easiest examples to check of a Calabi-Yau manifold is given by the Fermat quintic threefold, which is defined by the vanishing locus of the polynomial

f=x05+x15+x25+x35+x45{displaystyle f=x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3925c3889c915a849939ff753d50af969b22a1" aria-hidden="true" alt="{displaystyle f=x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}}" width="11908.4" height="1367.4">

Computing the partial derivatives of

f{displaystyle f}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" aria-hidden="true" alt="f" width="550.5" height="1080.4">$ gives the four polynomials

∂0f=5x04∂1f=5x14∂2f=5x24∂3f=5x34∂4f=5x44{displaystyle {begin{aligned}partial _{0}f=5x_{0}^{4}\partial _{1}f=5x_{1}^{4}\partial _{2}f=5x_{2}^{4}\partial _{3}f=5x_{3}^{4}\partial _{4}f=5x_{4}^{4}\end{aligned}}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cce98fefe2eb5637fb37103d83efacde92165e1" aria-hidden="true" alt="{displaystyle {begin{aligned}partial _{0}f=5x_{0}^{4}\partial _{1}f=5x_{1}^{4}\partial _{2}f=5x_{2}^{4}\partial _{3}f=5x_{3}^{4}\partial _{4}f=5x_{4}^{4}\end{aligned}}}" width="4720.4" height="7108.1">

Since the only points where they vanish is given by the coordinate axes in

P4{displaystyle mathbb {P} ^{4}}

[0:0:0:0:0]{displaystyle [0:0:0:0:0]}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eac07a25e3f98e75e0859f957d195943ed36d76e" aria-hidden="true" alt="{displaystyle [0:0:0:0:0]}" width="6395.7" height="1223.9">$ is not a point in

P4{displaystyle mathbb {P} ^{4}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef0f42e9d791c201ab1f11cfeaef6372d876364" aria-hidden="true" alt="{displaystyle mathbb {P} ^{4}}" width="1065.4" height="1152.1">$ .

As a Hodge Conjecture testbed[edit]

Another application of the quintic threefold is in the study of the infinitesimal generalized Hodge conjecture where this difficult problem can be solved in this case.^[2] In fact, all of the lines on this hypersurface can be found explicitly.

Dwork family of quintic three-folds[edit]

Another popular class of examples of quintic three-folds, studied in many contexts, is the Dwork family. One popular study of such a family is from Candelas, De La Ossa, Green, and Parkes,^[3] when they discovered mirror symmetry. This is given by the family^[4]^{pages 123-125}

fψ=x05+x15+x25+x35+x45−5ψx0x1x2x3x4{displaystyle f_{psi }=x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}-5psi x_{0}x_{1}x_{2}x_{3}x_{4}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4086448b26d70b4b5a31a90c3c60256ba037bf0e" aria-hidden="true" alt="{displaystyle f_{psi }=x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}-5psi x_{0}x_{1}x_{2}x_{3}x_{4}}" width="19916" height="1367.4">

where

ψ{displaystyle psi }

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" aria-hidden="true" alt="psi " width="651.5" height="1080.4">$ is a single parameter not equal to a 5-th root of unity. This can be found by computing the partial derivates of

fψ{displaystyle f_{psi }}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d654f8e8bbf8cc7b1ade54303c479bc06fda9781" aria-hidden="true" alt="{displaystyle f_{psi }}" width="1051.2" height="1223.9">$ and evaluating their zeros. The partial derivates are given by

∂0fψ=5x04−5ψx1x2x3x4∂1fψ=5x14−5ψx0x2x3x4∂2fψ=5x24−5ψx0x1x3x4∂3fψ=5x34−5ψx0x1x2x4∂4fψ=5x44−5ψx0x1x2x3{displaystyle {begin{aligned}partial _{0}f_{psi }=5x_{0}^{4}-5psi x_{1}x_{2}x_{3}x_{4}\partial _{1}f_{psi }=5x_{1}^{4}-5psi x_{0}x_{2}x_{3}x_{4}\partial _{2}f_{psi }=5x_{2}^{4}-5psi x_{0}x_{1}x_{3}x_{4}\partial _{3}f_{psi }=5x_{3}^{4}-5psi x_{0}x_{1}x_{2}x_{4}\partial _{4}f_{psi }=5x_{4}^{4}-5psi x_{0}x_{1}x_{2}x_{3}\end{aligned}}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6a1e40cb2c5cd91759eca3a76c5fb51de7a9a5" aria-hidden="true" alt="{displaystyle {begin{aligned}partial _{0}f_{psi }=5x_{0}^{4}-5psi x_{1}x_{2}x_{3}x_{4}\partial _{1}f_{psi }=5x_{1}^{4}-5psi x_{0}x_{2}x_{3}x_{4}\partial _{2}f_{psi }=5x_{2}^{4}-5psi x_{0}x_{1}x_{3}x_{4}\partial _{3}f_{psi }=5x_{3}^{4}-5psi x_{0}x_{1}x_{2}x_{4}\partial _{4}f_{psi }=5x_{4}^{4}-5psi x_{0}x_{1}x_{2}x_{3}\end{aligned}}}" width="11701.7" height="7108.1">

At a point where the partial derivatives are all zero, this gives the relation

xi5=ψx0x1x2x3x4{displaystyle x_{i}^{5}=psi x_{0}x_{1}x_{2}x_{3}x_{4}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a855a0f24627e8fd25e8ad0d8321fbbfbcdbe6" aria-hidden="true" alt="{displaystyle x_{i}^{5}=psi x_{0}x_{1}x_{2}x_{3}x_{4}}" width="8144" height="1367.4">$ . For example, in

∂0fψ{displaystyle partial _{0}f_{psi }}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6533277241f9c1eacae3e333ee518941decb726d" aria-hidden="true" alt="{displaystyle partial _{0}f_{psi }}" width="2036.6" height="1223.9">$ we get

5x04=5ψx1x2x3x4x04=ψx1x2x3x4x05=ψx0x1x2x3x4{displaystyle {begin{aligned}5x_{0}^{4}&=5psi x_{1}x_{2}x_{3}x_{4}\x_{0}^{4}&=psi x_{1}x_{2}x_{3}x_{4}\x_{0}^{5}&=psi x_{0}x_{1}x_{2}x_{3}x_{4}end{aligned}}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132cc71138f08e7fefbf6f452184491a2adacce0" aria-hidden="true" alt="{displaystyle {begin{aligned}5x_{0}^{4}&amp;=5psi x_{1}x_{2}x_{3}x_{4}\x_{0}^{4}&amp;=psi x_{1}x_{2}x_{3}x_{4}\x_{0}^{5}&amp;=psi x_{0}x_{1}x_{2}x_{3}x_{4}end{aligned}}}" width="8968" height="4237.8">

by dividing out the

5{displaystyle 5}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" aria-hidden="true" alt="5" width="500.5" height="936.9">$ and multiplying each side by

x0{displaystyle x_{0}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" aria-hidden="true" alt="x_{0}" width="1026.4" height="865.1">$ . From multiplying these families of equations

xi5=ψx0x1x2x3x4{displaystyle x_{i}^{5}=psi x_{0}x_{1}x_{2}x_{3}x_{4}}

∏xi5=ψ5∏xi5{displaystyle prod x_{i}^{5}=psi ^{5}prod x_{i}^{5}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5419e542fee244e819f720564183ceae9836e3d" aria-hidden="true" alt="{displaystyle prod x_{i}^{5}=psi ^{5}prod x_{i}^{5}}" width="7549.3" height="1654.5">

showing a solution is either given by an

xi=0{displaystyle x_{i}=0}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/491e215fe273840aab63acab32e73cb14f2fc8e8" aria-hidden="true" alt="x_{i}=0" width="2751.4" height="1080.4">$ or

ψ5=1{displaystyle psi ^{5}=1}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb65035ce83722fcaf9d05c073defe00d5e0541" aria-hidden="true" alt="{displaystyle psi ^{5}=1}" width="2940" height="1295.7">$ . But in the first case, these give a smooth sublocus since the varying term in

fψ{displaystyle f_{psi }}

ψ5=1{displaystyle psi ^{5}=1}

ψ{displaystyle psi }

[μ5a0:⋯:μ5a4]{displaystyle [mu _{5}^{a_{0}}:cdots :mu _{5}^{a_{4}}]} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f0030352e8def77a46367c7ab52a865bbc61f96" aria-hidden="true" alt="{displaystyle [mu _{5}^{a_{0}}:cdots :mu _{5}^{a_{4}}]}" width="6269.5" height="1367.4">

such that

μ5∑ai=ψ−1{displaystyle mu _{5}^{sum a_{i}}=psi ^{-1}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64400e650e414dbd39be7a9b0543bb9cf4d220e2" aria-hidden="true" alt="{displaystyle mu _{5}^{sum a_{i}}=psi ^{-1}}" width="5250.6" height="1582.7">

where

μ5=e2πi/5{displaystyle mu _{5}=e^{2pi i/5}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a120a559da01c7badb906bc92d872d43919104bd" aria-hidden="true" alt="{displaystyle mu _{5}=e^{2pi i/5}}" width="4669.5" height="1439.2">$ . For example, the point

[μ54:μ5−1:μ5−1:μ5−1:μ5−1]{displaystyle [mu _{5}^{4}:mu _{5}^{-1}:mu _{5}^{-1}:mu _{5}^{-1}:mu _{5}^{-1}]} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92741012079cf593b3c1bd629ad4920aeb88fa6" aria-hidden="true" alt="{displaystyle [mu _{5}^{4}:mu _{5}^{-1}:mu _{5}^{-1}:mu _{5}^{-1}:mu _{5}^{-1}]}" width="11382.2" height="1439.2">

is a solution of both

f1{displaystyle f_{1}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50dfd257a51e037112c917f8a9e47c9c053466df" aria-hidden="true" alt="f_{1}" width="944.4" height="1080.4">$ and its partial derivatives since

(μ5i)5=(μ55)i=1i=1{displaystyle (mu _{5}^{i})^{5}=(mu _{5}^{5})^{i}=1^{i}=1}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f5a0c234a8c1f11f4e49feeca7a57a016754ed" aria-hidden="true" alt="{displaystyle (mu _{5}^{i})^{5}=(mu _{5}^{5})^{i}=1^{i}=1}" width="9818.5" height="1439.2">$ , and

ψ=1{displaystyle psi =1}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96626cb8a694e730ead8946d1f396e1976b4b3ef" aria-hidden="true" alt="{displaystyle psi =1}" width="2486.1" height="1080.4">$ .

Other examples[edit]

Curves on a quintic threefold[edit]

Computing the number of rational curves of degree

1{displaystyle 1}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" aria-hidden="true" alt="1" width="500.5" height="936.9">$ can be computed explicitly using Schubert calculus. Let

T∗{displaystyle T^{*}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5426ed7abdea7d2ff995ad0e01bc4ca62a273855" aria-hidden="true" alt="T^{*}" width="910" height="400">$ be the rank

2{displaystyle 2}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" aria-hidden="true" alt="2" width="500.5" height="936.9">$ vector bundle on the Grassmannian

G(2,5){displaystyle G(2,5)}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da6e4105d4f7fe464603030f5a655dff4616ef1f" aria-hidden="true" alt="{displaystyle G(2,5)}" width="3011.7" height="1223.9">$ of

2{displaystyle 2}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" aria-hidden="true" alt="2" width="500.5" height="936.9">$ -planes in some rank

5{displaystyle 5}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" aria-hidden="true" alt="5" width="500.5" height="936.9">$ vector space. Projectivizing

G(2,5){displaystyle G(2,5)}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da6e4105d4f7fe464603030f5a655dff4616ef1f" aria-hidden="true" alt="{displaystyle G(2,5)}" width="3011.7" height="1223.9">$ to

G(1,4){displaystyle mathbb {G} (1,4)}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91d1c825a8a145c7201c0e2f75b7b9cdf760e50c" aria-hidden="true" alt="{displaystyle mathbb {G} (1,4)}" width="3003.7" height="1223.9">$ gives the projective grassmannian of degree 1 lines in

P4{displaystyle mathbb {P} ^{4}}

T∗{displaystyle T^{*}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5426ed7abdea7d2ff995ad0e01bc4ca62a273855" aria-hidden="true" alt="T^{*}" width="910" height="400">$ descends to a vector bundle on this projective Grassmannian. Its total chern class is

c(T∗)=1+σ1+σ1,1{displaystyle c(T^{*})=1+sigma _{1}+sigma _{1,1}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2a0ea12ecf2f252415c4ae21a5a453ff1374d99" aria-hidden="true" alt="{displaystyle c(T^{*})=1+sigma _{1}+sigma _{1,1}}" width="910" height="400">

in the Chow ring

A∙(G(1,4)){displaystyle A^{bullet }(mathbb {G} (1,4))}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f52155fae82bc08d335254d5439f71f10c1bce3" aria-hidden="true" alt="{displaystyle A^{bullet }(mathbb {G} (1,4))}" width="4987.1" height="1223.9">$ . Now, a section

l∈Γ(G(1,4),T∗){displaystyle lin Gamma (mathbb {G} (1,4),T^{*})}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2ea88d28c1e8bb50f3dfa11189f19a817fb7a1" aria-hidden="true" alt="{displaystyle lin Gamma (mathbb {G} (1,4),T^{*})}" width="7569.3" height="1223.9">$ of the bundle corresponds to a linear homogeneous polynomial,

l~∈Γ(P4,O(1)){displaystyle {tilde {l}}in Gamma (mathbb {P} ^{4},{mathcal {O}}(1))}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6b12e1abd229105939c1b1b399a0ce7b64539c" aria-hidden="true" alt="{displaystyle {tilde {l}}in Gamma (mathbb {P} ^{4},{mathcal {O}}(1))}" width="6797.9" height="1367.4">$ , so a section of

Sym5(T∗){displaystyle {text{Sym}}^{5}(T^{*})}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74bfa7936c1f50ab4de5435516458201381a5bac" aria-hidden="true" alt="{displaystyle {text{Sym}}^{5}(T^{*})}" width="4345.9" height="1367.4">$ corresponds to a quintic polynomial, a section of

Γ(P4,O(5)){displaystyle Gamma (mathbb {P} ^{4},{mathcal {O}}(5))}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8a713a6e2453d261cead6ee82a822463f5a915" aria-hidden="true" alt="{displaystyle Gamma (mathbb {P} ^{4},{mathcal {O}}(5))}" width="4991.1" height="1367.4">$ . Then, in order to calculate the number of lines on a generic quintic threefold, it suffices to compute the integral^[5]

∫G(1,4)c(Sym5(T∗))=2875{displaystyle int _{mathbb {G} (1,4)}c({text{Sym}}^{5}(T^{*}))=2875} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167a5a0ff6147a2cf8cd3181dee27bb18feff3da" aria-hidden="true" alt="{displaystyle int _{mathbb {G} (1,4)}c({text{Sym}}^{5}(T^{*}))=2875}" width="11723.6" height="2587.3">

This can be done by using the splitting principle. Since

c(T∗)=(1+α)(1+β)=1+(α+β)+αβ{displaystyle {begin{aligned}c(T^{*})&=(1+alpha )(1+beta )\&=1+(alpha +beta )+alpha beta end{aligned}}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082d1462d999424c34c3caf1273b866655dd5f5d" aria-hidden="true" alt="{displaystyle {begin{aligned}c(T^{*})&amp;=(1+alpha )(1+beta )\&amp;=1+(alpha +beta )+alpha beta end{aligned}}}" width="11440.9" height="2659.1">

and for a dimension

2{displaystyle 2}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" aria-hidden="true" alt="2" width="500.5" height="936.9">$ vector space,

V=V1⊕V2{displaystyle V=V_{1}oplus V_{2}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07883ac8dce55085bb4b2f7a3a40b208e965c062" aria-hidden="true" alt="{displaystyle V=V_{1}oplus V_{2}}" width="5401.3" height="1080.4">$ ,

Sym5(V)=⨁i=05(V1⊗5−i⊗V2⊗i){displaystyle {text{Sym}}^{5}(V)=bigoplus _{i=0}^{5}(V_{1}^{otimes 5-i}otimes V_{2}^{otimes i})} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e82d5b0a8013bbe6fc97cc25d9a9d483084fc3a" aria-hidden="true" alt="{displaystyle {text{Sym}}^{5}(V)=bigoplus _{i=0}^{5}(V_{1}^{otimes 5-i}otimes V_{2}^{otimes i})}" width="13113.1" height="3089.6">

so the total chern class of

Sym5(T∗){displaystyle {text{Sym}}^{5}(T^{*})}

c(Sym5(T∗))=∏i=05(1+(5−i)α+iβ){displaystyle c({text{Sym}}^{5}(T^{*}))=prod _{i=0}^{5}(1+(5-i)alpha +ibeta )} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec986694562f07cda7d4477af53050968f1ea46" aria-hidden="true" alt="{displaystyle c({text{Sym}}^{5}(T^{*}))=prod _{i=0}^{5}(1+(5-i)alpha +ibeta )}" width="16303.8" height="3161.4">

Then, the Euler class, or the top class is

5α(4α+β)(3α+2β)(2α+3β)(α+4β)5β{displaystyle 5alpha (4alpha +beta )(3alpha +2beta )(2alpha +3beta )(alpha +4beta )5beta } <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d41d76d5141a56fcedbb381c7d6e3ef52c3d39" aria-hidden="true" alt="{displaystyle 5alpha (4alpha +beta )(3alpha +2beta )(2alpha +3beta )(alpha +4beta )5beta }" width="18081.8" height="1223.9">

expanding this out in terms of the original chern classes gives

c6(Sym5(T∗))=25σ1,1(4σ12+9σ1,1)(6σ12+σ1,1)=(100σ2,2+225σ2,2)(6σ12+σ1,1)=325σ2,2(6σ12+σ1,1){displaystyle {begin{aligned}c_{6}({text{Sym}}^{5}(T^{*}))&=25sigma _{1,1}(4sigma _{1}^{2}+9sigma _{1,1})(6sigma _{1}^{2}+sigma _{1,1})\&=(100sigma _{2,2}+225sigma _{2,2})(6sigma _{1}^{2}+sigma _{1,1})\&=325sigma _{2,2}(6sigma _{1}^{2}+sigma _{1,1})end{aligned}}} <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b641f22837963f768ec648b3b663651529ae2a06" aria-hidden="true" alt="{displaystyle {begin{aligned}c_{6}({text{Sym}}^{5}(T^{*}))&amp;=25sigma _{1,1}(4sigma _{1}^{2}+9sigma _{1,1})(6sigma _{1}^{2}+sigma _{1,1})\&amp;=(100sigma _{2,2}+225sigma _{2,2})(6sigma _{1}^{2}+sigma _{1,1})\&amp;=325sigma _{2,2}(6sigma _{1}^{2}+sigma _{1,1})end{aligned}}}" width="20958.5" height="4237.8">

using the relations

σ1,1⋅σ12=σ2,2{displaystyle sigma _{1,1}cdot sigma _{1}^{2}=sigma _{2,2}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18ce5a7b1f9cf23368d0d91913aeab7cfac93b8e" aria-hidden="true" alt="{displaystyle sigma _{1,1}cdot sigma _{1}^{2}=sigma _{2,2}}" width="6236.2" height="1367.4">$ ,

σ1,12=σ2,2{displaystyle sigma _{1,1}^{2}=sigma _{2,2}}

$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b460dd77f94e0f37a207e5841e73a9d50bf44551" aria-hidden="true" alt="{displaystyle sigma _{1,1}^{2}=sigma _{2,2}}" width="4486.5" height="1510.9">$ .

Rational curves[edit]

Herbert Clemens (1984) conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. (Some smooth but non-generic quintic threefolds have infinite families of lines on them.) This was verified for degrees up to 7 by Sheldon Katz (1986) who also calculated the number 609250 of degree 2 rational curves.
Philip Candelas, Xenia C. de la Ossa, and Paul S. Green et al. (1991)
conjectured a general formula for the virtual number of rational curves of any degree, which was proved by Givental (1996) (the fact that the virtual number equals the actual number relies on confirmation of Clemens’ conjecture, currently known for degree at most 11 Cotterill (2012)).
The number of rational curves of various degrees on a generic quintic threefold is given by

2875, 609250, 317206375, 242467530000, …(sequence A076912 in the OEIS).

Since the generic quintic threefold is a Calabi–Yau threefold and the moduli space of rational curves of a given degree is a discrete, finite set (hence compact), these have well-defined Donaldson–Thomas invariants (the “virtual number of points”); at least for degree 1 and 2, these agree with the actual number of points.

References[edit]

^ Robbert Dijkgraaf (29 March 2015). “The Unreasonable Effectiveness of Quantum Physics in Modern Mathematics”. youtube.com. Trev M. Archived from the original on 2021-12-21. Retrieved 10 September 2015. see 29 minutes 57 seconds
^ Albano, Alberto; Katz, Sheldon (1991). “Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture”. Transactions of the American Mathematical Society. 324 (1): 353–368. doi:10.1090/S0002-9947-1991-1024767-6. ISSN 0002-9947.
^ Candelas, Philip; De La Ossa, Xenia C.; Green, Paul S.; Parkes, Linda (1991-07-29). “A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory”. Nuclear Physics B. 359 (1): 21–74. Bibcode:1991NuPhB.359…21C. doi:10.1016/0550-3213(91)90292-6. ISSN 0550-3213.
^ Gross, Mark; Huybrechts, Daniel; Joyce, Dominic (2003). Ellingsrud, Geir; Olson, Loren; Ranestad, Kristian; Stromme, Stein A. (eds.). Calabi-Yau Manifolds and Related Geometries: Lectures at a Summer School in Nordfjordeid, Norway, June 2001. Universitext. Berlin Heidelberg: Springer-Verlag. pp. 123–125. ISBN 978-3-540-44059-8.
^ Katz, Sheldon. Enumerative Geometry and String Theory. p. 108.

Arapura, Donu, “Computing Some Hodge Numbers” (PDF)
Candelas, Philip; de la Ossa, Xenia C.; Green, Paul S.; Parkes, Linda (1991), “A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory”, Nuclear Physics B, 359 (1): 21–74, Bibcode:1991NuPhB.359…21C, doi:10.1016/0550-3213(91)90292-6, MR 1115626
Clemens, Herbert (1984), “Some results about Abel-Jacobi mappings”, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton University Press, pp. 289–304, MR 0756858
Cotterill, Ethan (2012), “Rational curves of degree 11 on a general quintic 3-fold”, The Quarterly Journal of Mathematics, 63 (3): 539–568, doi:10.1093/qmath/har001, MR 2967162
Cox, David A.; Katz, Sheldon (1999), Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1059-0, MR 1677117
Givental, Alexander B. (1996), “Equivariant Gromov-Witten invariants”, International Mathematics Research Notices, 1996 (13): 613–663, doi:10.1155/S1073792896000414, MR 1408320
Katz, Sheldon (1986), “On the finiteness of rational curves on quintic threefolds”, Compositio Mathematica, 60 (2): 151–162, MR 0868135
Pandharipande, Rahul (1998), “Rational curves on hypersurfaces (after A. Givental)”, Astérisque, 1997/98 (252): 307–340, arXiv:math/9806133, Bibcode:1998math……6133P, MR 1685628

Quintic threefold – Wikipedia